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April 1996 Different clustering regimes in systems of hierarchically interacting diffusions
Achim Klenke
Ann. Probab. 24(2): 660-697 (April 1996). DOI: 10.1214/aop/1039639358

Abstract

We study a system of interacting diffusions $$\begin{eqnarray}dx_\xi(t)&=&\sum_{\zeta\in\Xi}a(\xi,\zeta)(x_\zeta(t)-x_\xi(t))dt \\ && + \sqrt{g(x_\xi(t))} dW_\xi(t) \qquad (\xi\in\Xi),\end{eqnarray}$$ indexed by the hierarchical group $\Xi$, as a genealogical two genotype model [where $x _\xi(t)$ denotes the frequency of, say, type A] with hierarchically determined degrees of relationship between colonies. In the case of short interaction range it is known that the system clusters. That is, locally one genotype dies out. We focus on the description of the different regimes of cluster growth which is shown to depend on the interaction kernel $a(\dot\quad,\dot\quad)$ via its recurrent potential kernel. One of these regimes will be further investigated by mean-field methods. For general interaction range we shall also relate the behavior of large finite systems, indexed by finite subsets $\Xi_n$ of, $\Xi$ to that of the infinite system. On the way we will establish relations between hitting times of random walks and their potentials.

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Achim Klenke. "Different clustering regimes in systems of hierarchically interacting diffusions." Ann. Probab. 24 (2) 660 - 697, April 1996. https://doi.org/10.1214/aop/1039639358

Information

Published: April 1996
First available in Project Euclid: 11 December 2002

zbMATH: 0862.60096
MathSciNet: MR1404524
Digital Object Identifier: 10.1214/aop/1039639358

Subjects:
Primary: 60K35

Rights: Copyright © 1996 Institute of Mathematical Statistics

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Vol.24 • No. 2 • April 1996
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